Groups saturated with finite simple groups $L_3(2^n)$ and $L_4(2^l)$

Let $\mathfrak{M}$ be a certain set of groups. For a group $G$, we denote by $\mathfrak{M}(G)$ the set of all subgroups of $G$ that are isomorphic to elements of $\mathfrak{M}$. A group $G$ is said to be saturated with groups from $\mathfrak{M}$ if any finite subgroup of $G$ is contained in some element of $\mathfrak{M}(G)$. We prove that if $G$ is a periodic group or a Shunkov group and $G$ is saturated with groups from the set $\{L_3(2^n), L_4(2^l)\mid n=1,2,\ldots; l=1,\ldots, l_0\},$ where $l_0$ is fixed, then the set of elements of finite order from $G$ forms a group isomorphic to one of the groups from the set $\{L_3 (R), L_4(2^l)\mid l=1,\ldots, l\}$, where $R$ is an appropriate locally finite field of characteristic $2$.